Solutions of a Derivative Nonlinear Schrödinger Hierarchy and Its Similarity Reduction

Symmetries in the third Painlevé equation arising from the modified Pohlmeyer-Lund-Regge hierarchy

“…We mention that our definition (2.5) differs from the previous paper [10]. Here we use not only the variable t i but alsot i .…”

“…The first couple of equations is the derivative nonlinear Schrödinger equation we have studied in [10], [11] and the second one is a modification of the coupled modified KdV equation.…”

“…This action was also introduced in ref. [10]. The relation of the independent variables are π(e φ(t,t) ) = e −φ(−t,−t) and π(q)(t,t) = −r(−t, −t), π(r)(t,t) = −q(−t, −t), π(q)(t,t) = −r(−t, −t), π(r)(t,t) = −q(−t, −t).…”

“…However, there have not been any satisfactory theories presented so far which could explain the relationship between Okamoto's theory, especially the symmetry of the affine Weyl group based on a Hamiltonian equation and the τ -function, and the soliton equations realized as representations of affine Lie algebras. So we develop the theory of the "modified" Pohlmeyer-Lund-Regge (mPLR) hierarchy that includes the derivative nonlinear Schrödinger hierarchy studied by Kakei and the author [10], [11], in the same way as the sine-Gordon hierarchy includes the mKdV hierarchy. The similarity reduction of the mPLR hierarchy gives the third Painlevé equation and its symmetry.…”

Drinfeld-Sokolov Hierarchies of Type A and Fourth Order Painleve Systems

“…The Drinfeld-Sokolov hierarchies are extensions of the KdV (or mKdV) hierarchy for the a‰ne Lie algebras [DS]. For type A ð1Þ n , they imply several Painlevé systems by similarity reductions [AS,KIK,KK1,KK2,NY1]; see Table 1. Such fact clarifies the origins of several properties of the Painlevé systems, Lax pairs, a‰ne Weyl group symmetries and particular solutions in terms of the Schur polynomials.…”